Skip to main content

Unitarily invariant norm inequalities involving Heron and Heinz means

Abstract

In this paper, we present some new inequalities for unitarily invariant norms involving Heron and Heinz means for matrices, which generalize the result of Theorem 2.1 (Fu and He in J. Math. Inequal. 7(4):727-737, 2013) and refine the inequality of Theorem 6 (Zhan in SIAM J. Matrix Anal. Appl. 20: 466-470, 1998). Our results are a refinement and a generalization of some existing inequalities.

MSC:47A30, 15A60.

1 Introduction

Throughout, let M m , n be the space of m×n complex matrices and M n = M n , n .

A norm is called unitarily invariant norm if UAV=A for all A M n and for all unitary matrices U,V M n . Two classes of unitarily invariant norms are especially important. The first is the class of the Ky Fan k-norm ( k ) , defined as

A ( k ) = j = 1 k s j (A),k=1,,n,

where s i (A) (i=1,,n) are the singular values of A with s 1 (A) s n (A), which are the eigenvalues of the positive semidefinite matrix |A|= ( A A ) 1 2 , arranged in decreasing order and repeated according to multiplicity. The second is the class of the Schatten p-norm ( p ) , defined as

A p = ( j = 1 n s j p ( A ) ) 1 p = ( tr | A | p ) 1 p ,1p<.

For two nonnegative real numbers a and b, the Heinz mean and Heron mean in the parameter v, 0v1, are defined, respectively, as

H v ( a , b ) = a v b 1 v + a 1 v b v 2 , F α ( a , b ) = ( 1 α ) a b + α a + b 2 .

Note that H 0 (a,b)= H 1 (a,b)= a + b 2 (the arithmetic mean of a and b) and H 1 2 (a,b)= a b (the geometric mean of a and b). It is easy to see that as a function of v, H v (a,b) is convex, attains its minimum at v= 1 2 , and attains its maximum at v=0 and v=1.

The operator version of the Heinz mean [1] asserts that if A, B and X are operators on a complex separable Hilbert space such that A and B are positive, then for every unitarily invariant norm , the function g(v)= A v X B 1 v + A 1 v X B v is convex on [0,1], attains its minimum at v= 1 2 , and attains its maximum at v=0 and v=1. Moreover, the operator version of the Heron mean [2] asserts that f(α)=(1α) A 1 2 X B 1 2 +α( A X + X B 2 ).

Let A,B,X M n , A, B are positive definite, Kaur and Singh [2] have proved the following inequalities for any unitarily invariant norm :

1 2 A v X B 1 v + A 1 v X B v ( 1 α ) A 1 2 X B 1 2 + α ( A X + X B 2 )
(1.1)

and

A 1 2 X B 1 2 1 2 A 2 3 X B 1 3 + A 1 3 X B 2 3 1 2 + t A X + t A 1 2 X B 1 2 + X B ,
(1.2)

where 1 4 v 3 4 , α[ 1 2 ,) and t(2,2].

Replacing A, B by A 2 , B 2 in (1.1) and (1.2), then putting u=2v, the following inequalities hold:

1 2 A u X B 2 u + A 2 u X B u ( 1 α ) A X B + α ( A 2 X + X B 2 2 )
(1.3)

and

AXB 1 2 A 4 3 X B 2 3 + A 2 3 X B 4 3 1 t + 2 A 2 X + t A X B + X B 2 ,
(1.4)

where 1 2 u 3 2 , α[ 1 2 ,) and t(2,2].

Zhan proved in [3] that if A,B,X M n , such that A, B are positive semidefinite, then

A u X B 2 u + A 2 u X B u 2 t + 2 A 2 X + t A X B + X B 2
(1.5)

for 1 2 u 3 2 and t(2,2].

Let A,B,X M n , such that A, B are positive semidefinite, for 1 2 u 3 2 and t(2,2]; Fu et al. in [4] proved that

2 A X B + 2 ( 1 2 3 2 A r X B 2 r + A 2 r X B r d r 2 A X B ) 2 t + 2 A 2 X + t A X B + X B 2 .
(1.6)

Recently, Kaur et al. [5], He et al. [6] and Bakherad et al. [7] have studied similar topics.

For the sake of convenience, we set

g(u)= A u X B 2 u + A 2 u X B u 2 .

In Section 2, we will generalize and refine some existing inequalities for unitarily invariant norms involving Heron and Heinz means for matrices and present some new refinements of the inequalities above.

2 Main results

In this section, we firstly utilize the convexity of the function g(u) to obtain a unitarily invariant norms inequality that leads to another version of the inequality (1.6), which is also the refinement of the inequality (1.5).

To obtain the results, we need the following lemma on convex functions [8, 9].

Lemma 2.1 Let f be a real valued continuous convex function on an interval [a,b] which contains ( x 1 , x 2 ). Then for x 1 x x 2 , we have

f(x) f ( x 2 ) f ( x 1 ) x 2 x 1 x x 1 f ( x 2 ) x 2 f ( x 1 ) x 2 x 1 .

Theorem 2.2 Let A,B,X M n , such that A, B are positive semidefinite. Then for any unitarily invariant norm , 1 2 u 3 2 and α[ 1 2 ,),

A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) ,
(2.1)

where r 0 =min[ u 2 ,1 u 2 ].

Proof For 1 2 u1, by the convexity of the function g(u) and Lemma 2.1, presented above, we have

g(u) g ( 1 ) g ( 1 2 ) 1 2 u 1 2 g ( 1 ) g ( 1 2 ) ) 1 2 ,

which implies

g(u)2(1u)g ( 1 2 ) +(2u1)g(1).
(2.2)

By (1.3) and (2.2), we have

A u X B 2 u + A 2 u X B u 4(1u) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) +2(2u1)AXB.

So,

A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .
(2.3)

For 1u 3 2 , by the convexity of the function g(u) and Lemma 2.1, presented above, we have

g(u) g ( 3 2 ) g ( 1 ) 1 2 u g ( 3 2 ) 3 2 g ( 1 ) 1 2 ,

which implies

g(u)(32u)g(1)+2(u1)g ( 3 2 ) .
(2.4)

By (1.3) and (2.4), we have

A u X B 2 u + A 2 u X B u 2 ( 3 2 u ) A X B + 4 ( u 1 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .

So,

A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .
(2.5)

By (2.3) and (2.5), for 1 2 u 3 2 , α[ 1 2 ,) and r 0 =min[ u 2 ,1 u 2 ], we have the following equivalent inequality:

A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .

The proof is completed. □

Remark 2.3 With a simple computation between the upper bounds in (1.3) and (2.1), obviously we have

( 1 α ) A X B + α ( A 2 X + X B 2 2 ) ( 4 r 0 1 ) A X B 2 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) = ( 4 r 0 1 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) ( 4 r 0 1 ) A X B = ( 4 r 0 1 ) ( ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) A X B ) > 0 .

Thus the inequality (2.1) is a refinement of the inequality (1.3).

Now, we present a refinement of the inequality AXB(1α)AXB+α( A 2 X + X B 2 2 ).

Theorem 2.4 Let A,B,X M n , such that A, B are positive semidefinite. Then for any unitarily invariant norm , 1 2 u 3 2 , and α[ 1 2 ,), we have

2 A X B + 2 ( 1 2 3 2 A u X B 2 u + A 2 u X B u d u 2 A X B ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) ,
(2.6)

where r 0 =min[ u 2 ,1 u 2 ].

Proof For 1 2 u1, from Theorem 2.2, we have

A u X B 2 u + A 2 u X B u 4 ( 1 u ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) + 2 ( 2 u 1 ) A X B .

By integrating both sides of the inequality above, we have

1 2 1 A u X B 2 u + A 2 u X B u d u 4 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) 1 2 1 ( 1 u ) d u + 2 A X B 1 2 1 ( 2 u 1 ) d u = 1 2 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) + 1 2 A X B .
(2.7)

For 1u 3 2 , from Theorem 2.2, we have

A u X B 2 u + A 2 u X B u 2 ( 3 2 u ) A X B + 4 ( u 1 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .

Similarly, by integrating both sides of the inequality above, we have

1 3 2 A u X B 2 u + A 2 u X B u d u 2 A X B 1 3 2 ( 3 2 u ) d u + 4 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) 1 3 2 ( 2 u 1 ) d u = 1 2 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) + 1 2 A X B .
(2.8)

It follows from (2.7) and (2.8) that

1 2 3 2 A u X B 2 u + A 2 u X B u du ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) +AXB,

which is equivalent to

2 A X B + 2 ( 1 2 3 2 A u X B 2 u + A 2 u X B u d u 2 A X B ) 2 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .

The proof is completed. □

Remark 2.5 Obviously,

1 2 3 2 A u X B 2 u + A 2 u X B u du2AXB0.

Thus, the inequality (2.6) is a refinement of the inequality AXB(1α)AXB+α( A 2 X + X B 2 2 ).

Taking α= 2 t + 2 (2<t2), the following corollaries are obtained.

Corollary 2.6 Let A,B,X M n , such that A, B are positive semidefinite. Then for any unitarily invariant norm and 1 2 u 3 2 ,

A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) t + 2 A 2 X + t A X B + X B 2 ,
(2.9)

where r 0 =min[ u 2 ,1 u 2 ] and 2<t2.

Corollary 2.7 Let A,B,X M n , such that A, B are positive semidefinite. Then for any unitarily invariant norm , 1 2 u 3 2 ,

2 A X B + 2 ( 1 2 3 2 A u X B 2 u + A 2 u X B u d u 2 A X B ) 2 t + 2 A 2 X + t A X B + X B 2 ,
(2.10)

where r 0 =min[ u 2 ,1 u 2 ] and 2<t2.

Thus, on the one hand, the inequality (2.9) is a refinement of the inequality AXB 1 t + 2 A 2 X+tAXB+X B 2 , and also another version of the inequality (1.6); on the other hand, the inequality (2.10) is just the inequality proved in [4], so the inequality (2.6) presented in Theorem 2.4 is also the generalization of the inequality proved in [4].

References

  1. Kittaneh F: On the convexity of the Heinz means. Integral Equ. Oper. Theory 2010, 68: 519–527. 10.1007/s00020-010-1807-6

    MathSciNet  Article  MATH  Google Scholar 

  2. Kaur R, Singh M: Complete interpolation of matrix versions of Heron and Heinz means. Math. Inequal. Appl. 2013, 16: 93–99.

    MathSciNet  MATH  Google Scholar 

  3. Zhan X: Inequalities for unitarily invariant norms. SIAM J. Matrix Anal. Appl. 1998, 20: 466–470. 10.1137/S0895479898323823

    Article  MATH  MathSciNet  Google Scholar 

  4. Fu X, He C: On some inequalities for unitarily invariant norms. J. Math. Inequal. 2013,7(4):727–737.

    MathSciNet  Article  MATH  Google Scholar 

  5. Kaur R, Moslehian MS, Singh M, Conde C: Further refinements of the Heinz inequality. Linear Algebra Appl. 2014, 447: 26–37.

    MathSciNet  Article  MATH  Google Scholar 

  6. He CJ, Zou LM, Qaisar S: On improved arithmetric-geometric mean and Heinz inequalities for matrices. J. Math. Inequal. 2012,6(3):453–459.

    MathSciNet  Article  MATH  Google Scholar 

  7. Bakherad M, Moslehian MS: Reverses and variations of Heinz inequality. Linear Multilinear Algebra 2014. 10.1080/03081087.2014.880433

    Google Scholar 

  8. Bhatia R, Sharma R: Some inequalities for positive linear maps. Linear Algebra Appl. 2012, 436: 1562–1571. 10.1016/j.laa.2010.09.038

    MathSciNet  Article  MATH  Google Scholar 

  9. Wang S, Zou L, Jiang Y: Some inequalities for unitarily invariant norms of matrices. J. Inequal. Appl. 2011., 2011: Article ID 10

    Google Scholar 

Download references

Acknowledgements

The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Haisong Cao.

Additional information

Competing interests

The authors declare that they have no competing interests. HC is responsible for all the whole of the article appearing.

Authors’ contributions

JW carried out the matrix operator theory studies and participated in the conception and design. HC conceived of the study, participated in its design and drafting the manuscript. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cao, H., Wu, J. Unitarily invariant norm inequalities involving Heron and Heinz means. J Inequal Appl 2014, 288 (2014). https://doi.org/10.1186/1029-242X-2014-288

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-288

Keywords

  • unitarily invariant norm
  • positive definite matrices
  • convex function
  • Heron means
  • Heinz means